Programs Does a Mathematics PhD take special talent?

I started loving mathematics ever since my freshman year in High School, and I was even considered one of the brightest students in class because I normally studied ahead the others (they'd be in one chapter, and I'd be a chapter or two ahead.) Now that I'm in college, my Precalculus professor gave me a taste of what the world of mathematics is like at the university level, and it wasn't pretty. Still, even though I'm thinking of majoring in Business or Chemistry, I'm thinking of challenging myself and to aim for a master's or a PhD in mathematics. My question is: does it take special talent to earn a master's or a PhD in Mathematics? Does it take a lot more than just motivation and taking all the possible mathematics courses out there (Real/Complex Analysis, Number Theory, Ordinary/Partial Differential Equations, etc.?) And finally, is it true that in order to earn a PhD, I'll have to make some sort of special mathematical discovery?

In order to get a PhD you are required to do research in a particular field and write papers about what you've discovered. However these discoveries in general aren't that shocking as you might think now.

A PhD does require talent. You certainly need good grades to get accepted for a PhD (or graduate school).

And finally, is it true that in order to earn a PhD, I'll have to make some sort of special mathematical discovery?

This is precisely what a PhD is. A PhD is the statement that you have made some original contribution to human knowledge on the topic of your thesis. Now, normally this contribution is very small and specialized, such that perhaps only a handful of people in the world might truly understand your result.

If you just want to continue studying mathematics past undergraduate, but still in classes, perhaps with only a small research project, this is what a masters is.

You should probably take mathematics courses at the undergraduate level before you even begin worrying about it though.

...my Precalculus professor gave me a taste of what the world of mathematics is like at the university level, and it wasn't pretty.

Can you elaborate on what this means? Why wasn't it pretty?

There is a point at which you have to make a transition from computational mathematics (high-school algebra, calculus, etc.) to abstract mathematics (analysis, abstract algebra, etc.). This does not have to be a bad thing, though.

...my Precalculus professor gave me a taste of what the world of mathematics is like at the university level, and it wasn't pretty.

Can you elaborate on what this means? Why wasn't it pretty?

Sure. I asked him what mathematics will be like at the university level, and he showed me an example of what Caculus 2 and Partial Differential Equations will be like, and I got lost during the first minute of his example.

I believe that one minute isn't enough to tell how beautiful mathematics is and how profound the idea behind those ugly equations is

I don't major in math or science, but I think being the brightest in class (or being the best at math in class) is not the right reason to go for math PhD. I believe it would be some sort of strong motivation inside (of course, plus talent). It has to be very very strong, I think.

I didn't doubt your talent or your passion. Just that I think you should prepare yourself well (mentally!) before deciding on doing math. Personally I think doing PhD in math/science means not-reaching-PhD-in-math/science. It means giving yourself an excuse to spend more time of your life living with math.
But what do I know?!

I guess for right now, I'll aim for a degree in Business Finance or Chemistry, and then when I feel as if I'm ready both physically and mentally, I'll challenge myself and go back to school to further continue my studies in Mathematics.

When you don't understand the mathematics it often comes off as overwhelming gibberish, your experience of it 'not being pretty' is not uncommon. The process of buckling down and beginning to 'grok' a new piece of mathematics can also be a painful experience. But once you have understood it the math will often come off as elegant, beautiful and believe it or not quite simple, and you will begin to marvel at how sensible and straightforward it was for mathematicians to construct the mathematics the way they did.

As far as requiring a special talent to do advanced mathematics, that is difficult to say.. In the sense that you're using the word 'talent', i.e. like some inborn spark, I would say no. But cerebrally mathematics is a 'processor intensive' activity. To do advanced mathematics you will need a number of interrelated talents which can be developed, mostly by doing mathematics, as well as by studying related fields such as philosophy and computer science.

Thus since you will have to do a lot of mathematics in order to develop the skills required to progress, I would recommend that if you love mathematics more than business finance or chemistry, than you should choose mathematics.

Keep in mind that neither business or chemistry require much math past calculus, if that, and that means you won't be able to apply for a masters/PhD in math down the road without going back and taking a significant amount of undergraduate math courses. And a PhD doesn't just mean you're good at math classes, it means you can come up with something original and spend years working on it, just to see it not work out in the end. A few of my friends did PhDs in math - in both cases, their PhD work was less than 10 months of their 3-5 year PhD. The first few years were projects that just didn't pan out, and in math, you don't write papers about things that didn't work.

Sure. I asked him what mathematics will be like at the university level, and he showed me an example of what Caculus 2 and Partial Differential Equations will be like, and I got lost during the first minute of his example.

If I showed you a page of text in a foreign language, say Russian or Japanese (assuming you don't speak either), would you have any hope of getting past "the first minute?" Mathematics is a highly developed language, if you haven't learnt it, you can't read it. Pretty simple.

If you do pursue mathematics, Calc2 is very basic material that should eventually become just like speaking English. Yes, you may occasionally spell something wrong or make an awkward sentence but, if you didn't speak English, it would take some practice to get to the point that you take for granted right now.

There is sometimes a strange idea that, if you are going to be a good mathematician, you should automatically understand everything everyone has ever written without having to study (yes, I exaggerate, but only a little). Everyone has to work. Some more than others, but if you enjoy math and are willing to put in the effort, it is within your reach.

If you grab a random paper off of Arxiv that isn't in the overall field of a particular mathematician, chances are that they will struggle too.

If you're at the level of precalculus, its only natural to be baffled by PDE's and calc 2. Those are at least 4-6 semester courses ahead of you! You'd need to know about ODE's, which requires reasonable mastery of single variable calculus, which requires extensive knowledge of precalculus... etc.

And by the way how are you taking precalculus at college level? I've never heard of a school not imparting it or an equivalent (precal/trig) before 10th grade.

Yea I agree, it looks scary because you don't understand it yet. I used to think calculus was some high level mathematics that is incredibly hard to understand. I was scared when I found out I had to take 3 semester of it.

Now I've taken all of them, loved them and aced them with no problems.

Most universities require chem majors to take alot of math classes as well. I know at my school they go up to Calc III with ODE and Linear Algebra as recommended electives. So if you go that route, you'll have some extra time to decide.

in answer to the title question, i think it does, but the only way to find out if you have that talent is to try to get one. and not succeeding at first does not necessarily mean you don't. it also requires desire, perserverance, opportunity, good teachers, money.

Bogrune, I think it does take talent, but more than anything, it takes consistency and sheer determination. I thought I wanted a masters in math, and started the classes and failed miserably because I just wasn't motivated anymore. I needed something more applied so that is why I am now pursuing physics grad school. I think to go all the way until a PhD you definitely need to know that that is what you want to do the rest of your life, otherwise you will fail- just like if you go to med school to become a doctor or law school to become a lawyer just for the money and not because you love medicine or law.

And by the way how are you taking precalculus at college level? I've never heard of a school not imparting it or an equivalent (precal/trig) before 10th grade.

Well what do you expect? I live in California, and last I've heard, the educatoinal standards here are pretty low! :)

I am very determined when it comes to Mathematics. Unfortunately, I have to take Precalculus for a third time because I passed with a "D", and I needed to earn a "C" so that I may progress to Calculus I, and though I have to take it for a third time, I'm still not giving up. I may decide to major for a degree in Finance or Economics, and I was told that the highest level of mathematics I need for it (can't remember if it was Finance or Economics) is Linear Algebra and Differential Equations, and as for Chemical Engineering, I may need to progress even further.

Anyway, though I find mathematics very difficult at times (normally when it comes to Vectors or Analytic Geometry), I don't give up. What I love about Mathematics is the analyzation it takes, which I usually find quite fun to work with!

It is very good that you enjoy math! That is probably the most important part of getting a math PhD.

Although you aren't necessarily running far behind, it is slightly worrying that you are getting Ds or failing courses like precalculus. As you progress through mathematics, concepts become more abstract and therefore harder. It is important to develop good habits early on, like being timely with homework, identifying areas that are giving you trouble early on, and, very importantly, if you cannot figure something out talk to the professor as soon as possible!

Most graduate programs require/expect at least some background in analysis and/or abstract algebra, so as long as you don't need to retake another critical course, you should be able to get these in senior year.

Well what do you expect? I live in California, and last I've heard, the educatoinal standards here are pretty low! :)

I am very determined when it comes to Mathematics. Unfortunately, I have to take Precalculus for a third time because I passed with a "D", and I needed to earn a "C" so that I may progress to Calculus I, and though I have to take it for a third time, I'm still not giving up. I may decide to major for a degree in Finance or Economics, and I was told that the highest level of mathematics I need for it (can't remember if it was Finance or Economics) is Linear Algebra and Differential Equations, and as for Chemical Engineering, I may need to progress even further.

Anyway, though I find mathematics very difficult at times (normally when it comes to Vectors or Analytic Geometry), I don't give up. What I love about Mathematics is the analyzation it takes, which I usually find quite fun to work with!

I share Bogrune's sentiments, you should really try to learn precalculus thoroughly because you'll be seeing and using its contents again and again throughout a variety of math and physics courses. Some words of reassurance: I think university level calculus/analysis are probably the toughest undergrad math courses, but after you get through those, material like ODE's, PDE's, and integral transforms will seem VERY easy in comparison.